A=Expand[(((f-a-b x-c (x^2)))^2)]

(* 
Take the expectation of the above keeping product terms separate.  In the above "(x^i)" is the expectation of x or the raw moment of x.
*)
expectationOfA=(a^2)-2 a f+f2+2 a b x-2 b fx+(b^2) x2+2 a c x2-2 c fx2+2 b c x3+(c^2) x4

(* lets take the partial derivatives with respect to a,b,c set the result equal to zero and solve for a,b,c *)
eq1=D[expectationOfA,a]
eq2=D[expectationOfA,b]
eq3=D[expectationOfA,c]

Solve[eq1==0,{a}]

(* put this value of a into the previous two equations *)
eq2prime=eq2/.{a->f-b x -c x2}//Simplify
eq3prime=eq3/.{a->f-b x -c x2}//Simplify

(* These two remaining equations for b,c *)
sol=Solve[{eq2prime==0,eq3prime==0},{b,c}]//Simplify

(* Lets convert the raw moments into central moments.  We denote the mean of x by x itself *)
Simplify[sol/.{x2->m2+(x^2),x3->m3+3 m2 x+(x^3),x4->m4+4m3 x + 6 m2 (x^2)+(x^4)}]